The Effect of Vector Control Strategy against Dengue Transmission between Mosquitoes and Human

With the consideration of the mechanism of prevention and control for the spread of dengue fever, a mathematical model of dengue fever dynamical transmission between mosquitoes and humans, incorporating a vector control strategy of impulsive culling of mosquitoes, is proposed in this paper. By using the comparison principle, Flo-quet theory and some of analytical methods, we obtain the basic reproductive number R 0 for this infectious disease, which illustrates the stability of the disease-free periodic solution and the uniform persistence of the disease. Further, the explicit conditions determining the backward or forward bifurcation are obtained and we show that the culling rate φ has a major effect on the occurrence of backward bifurcation. Finally, numerical simulations are given to verify the correctness of theoretical results and the highest efficiency of vector control strategy.


Introduction
Dengue fever is a fast emerging pandemic-prone viral disease in many parts of the world, which was first discovered in Cairo of Egypt, Indonesia-Jakarta and Philadelphia 1779 and was named arthritis fever and break-bone fever according to clinical symptoms [8,18].Dengue fever is a re-emergent disease affecting people in more than 100 countries in the tropical and subtropical areas.The symptoms of the disease are characterized by high fever, frontal headache, pain behind the eyes, joint pains, nausea, vomiting etc [27,33].Every year, 500,000 cases reports of dengue are received by the World Health Organization (WHO), with more than 2.5 billion people at risk.It is well know that dengue fever is a mosquito-borne infectious disease, the female mosquitoes of Aedes aegypti and Aedes albopictus are the prominent carriers of dengue fever virus of Flaviviridae family [7,33].The infected mosquitoes transfer infection on biting susceptible persons, and then susceptible mosquito bites an infected person, it gets infected.
The control and hence eradication of infectious diseases is one of the major concerns in the study of mathematical epidemiology [1,2,11,21,31] and the references therein.In the past nearly Corresponding author.Email: lfnie@163.comtwenty years, the prevention and control of the spread of dengue fever (or other vector-borne diseases) are also caused extensive attention of many scholars (see, for instance, [5,12,13,24] and the references therein).Especially, Esteva et al. [24] proposed a host-vector dengue fever model with varying total population size and obtained the global asymptotic stability of equilibria for this model.Derouich et al. [10] introduced a mathematical model to simulate the succession of dengue with variable human populations, and analysed the stability of equilibria for this model.Garba et al. [16] structured a basic single strain dengue model, which incorporates the dynamics of exposed humans and vectors, and obtained the basic reproduction number and the phenomenon of backward bifurcation, where the stable disease-free equilibrium coexists with a stable endemic equilibrium.Further, they also extended the model to incorporate an imperfect vaccine against the spread of dengue.For more research results also can be found in [3,9,14,15,19,[28][29][30] and the references therein.
Dengue vaccination research and development are began in 1940s, but no specific vaccine and treatment are available for this vector-borne disease in recent years due to the limited appreciation of global disease burden and the potential markets.Therefore, how to prevent, control and put an end to the spread of dengue fever have always been hot issues in medicine and mathematical epidemiology.Considering mosquitoes are the principal vectors in spread of the disease, hence vector population control through insecticides is the possible ways to prevent human population from dengue fever virus.Based on this idea, significant advances were made by Macdonald [25] who proposed that the most effective control strategy against vector-borne infections is to kill adult mosquitoes.From a theoretical perspective, many mathematical models have been developed to describe the control of vector population.Examples can be found in [26] where they studied the effects of awareness and mosquito control on dengue fever, and indicated that the sufficiently large amount of vector control is required to control the disease.And in [4], Amaku et al. considered the impact of vector-control strategies on the human prevalence of dengue.Other examples also can be found in [17,20] and the references therein, to just a few.
A noticed fact is that most of above mathematical models considering control measures on vector population invariably assume that the pesticides affect vector population continuously.Generally, however, the culling of vector population is usually put in practice at discrete certain times.So for this reason, Xu et al. [34] presented a mathematical model to describe the transmission of West Nile virus between vector mosquitoes and birds, incorporating a control strategy of culling mosquitoes and defined by impulsive differential equations.
Motivated by these facts, we propose, in this paper, a mathematical model of dengue with saturation and bilinear incidence, and impulsive culling of mosquitoes.The dynamical behaviour of this model was investigated and the influence of impulsive vector control for the preventing of the disease was discussed.The rest of the paper is structured as follows.We formulate a mathematical model and give some useful lemmas in Section 2. Some threshold value conditions of the disease-free periodic solution are presented in Section 3 to control the disease.We discuss the uniform persistence of the disease in Section 4 and also provide a explicit conditions determining the backward or forward bifurcation in Section 5. Finally, we give the numerical simulation and discussion in the last section.

Model formulation and preliminaries
In this section, we propose a mathematical model of dengue fever with impulsive culling mosquitoes.For convenience, we separate human population into three classes: susceptible S h , infectious I h , recovered R h , and female mosquitoes are divided into two classes: uninfected S m , infectious I m .For establishing this model, we come up with the following assumptions.
(A 1 ) The human population is recruited at the rate of µ h K, K is the maximum size of people and µ h is natural death rate, and because dengue fever also causes mortality in humans, we assume that d h is the disease-induced death rate for humans, γ h is the recovery rate of humans, N h is the total size of human population.
(A 2 ) Assuming that Λ m is the recruitment rate of mosquitoes, N m is the total number of mosquitoes and µ m is the natural birth/death rate of mosquitoes.
(A 3 ) Average biting rate of mosquitoes is b, ρ hm and ρ mh are the transmission probabilities from human to mosquitoes and from mosquitoes to human respectively.We assume that the incidence of infected mosquito to susceptible humans is the saturation incidence due to the "psychological" effect or the inhibition effect, and assume that the incidence of infected humans to susceptible mosquito is bilinear incidence since it is not impacted by these issues.They are given by bρ hm I m (t)S h (t) respectively, where α is positively constant.
(A 4 ) We suppose that impulsive culling of infectious and susceptible mosquitoes happens at a rate φ.
Based on the above assumptions, we have the following mathematical model with impulsive control From the first to fifth equations of model (2.1), we obtain that the total numbers of humans and vectors at time t satisfy Obviously, from the above two equations, it is easy to get ρ mh Transmission rate of human to mosquito (day −1 ) 5.6478e −5 ∼ 7.3135e −4 Pandey [28] µ h The natural death rate of human (day) 1/25000 Garba S.M. [16] µ m The natural death rate of mosquito (day −1 ) 0.0378 ∼ 0.0781 Pandey [28] d h The diseased death rate of human (day −1 ) 1e −3 Amaku M. [4] γ h The rate of recovery in human (day −1 ) 0.1521 ∼ 0.4440 Pandey [28] φ The culling rate for mosquitoes (day −1 ) 0 ∼ 1 Estimate Therefore, from biological considerations, we only need to analyze the dynamical behavior of model (2.1) in the region Thus, ν(t) is a positive ω-periodic function.So, x(t) = e µt ν(t) is a solution of system (2.3).This completes the proof.
The following Lemma 2.2 is from [23], although very simple, but very useful.
Lemma 2.2.Consider the following impulsive differential equation where a, b, c, d are positive constants and p ∈ (0, 1).Then equation (2.4) has a unique positive periodic solution ( x 1 (t), x 2 (t)) which is globally asymptotically stable, where

Stability of the disease-free periodic solution
In this section, we discuss the existence and stability of the disease-free periodic solution of model (2.1), in which infected individuals are completely absent.That is, I h (t) = 0 and I m (t) = 0.In this case, model (2.1) reduces to the following subsystem Therefore, model (2.1) admits a unique disease-free periodic solution ( S h (t), 0, 0, S m (t), 0).For discussion of the stability of disease-free periodic solution of model (2.1), we define the following matrix functions Let A(t) be an n × n matrix function, Φ A(•) (t) be the fundamental solution matrix of the linear ordinary differential system dX(t)/dt = A(t)X(t), and ρ(Φ A(•) (T)) be the spectral radius of where and Obviously, the monodromy matrix of model (3.3) equals where * stands for a non-zero block matrix.Further, the Floquet multiplier of model (2.1) is the eigenvalue of ρ(P 1 Φ F−V (T)) and ρ(Φ M (T)).We can easily get the following theorem if 1) admits a disease-free periodic solution ( S h (t), 0, 0, S m (t), 0) which is locally asymptotically stable.
The following Theorem 3.2 is on the global asymptotic stability of the disease-free periodic solution of model (2.1).Theorem 3.2.If R 0 < 1 holds, then the disease-free periodic solution ( S h (t), 0, 0, S m (t), 0) of model (2.1) is globally asymptotically stable.
Proof.From Theorem 3.1, we have verified the local asymptotic stability of the disease-free periodic solution of model (2.1).So we only need to prove the global attractivity of this solution.By R 0 From the first, fourth, sixth and ninth equations of model (2.1), one has

Consider the auxiliary system
By Lemma 2.2, model (3.4) has a unique positively periodic solution ( ω 1 (t), ω 2 (t)) which is globally asymptotically stable, where By the comparison theorem of impulsive differential equations (more detail see Lakshmikantham et al. [6,23]), there exists an integer n 1 > 0 such that That is, Further, from (3.5) and the second, fifth, seventh and tenth equations of model (2.1), we can get

Now, we consider the following auxiliary system
where From these, for any 1 > 0, there exists an integer n 2 ≥ n 1 , such that I h (t) < 1 and I m (t) < 1 for t ≥ n 2 T. From the first, fourth, sixth and ninth equations of model (2.1), it can be easily shown

Consider the auxiliary system
Then by the comparison theorem, there exists a integer n 3 ≥ n 2 such that Since 1 is arbitrarily small, it follows from the above inequality and (3.5) that z 1 (t) Finally, from the second equation of model (2.1), we have lim t→+∞ R h (t) = 0. From this and (3.7), (3.9), we have that the disease-free periodic solution of model (2.1) is globally attractive.The proof is complete.

Uniform persistence of the disease
In this section, we turn to the uniform persistence of the disease for model (2.1).
we can chose small enough positive constants η, 1 and 2 such that where and ( S h (t), S m (t)) are given by (3.2).Firstly, we prove lim sup Otherwise, there exists a t 1 > 0 such that I h (t) < η or I m (t) < η for all t ≥ t 1 .Without loss generality, we suppose that I h (t) < η and I m (t) < η for all t ≥ t 1 .By the first, fourth, sixth and ninth equations of model (2.1), we have

Consider the auxiliary system
By Lemma 2.2, system (4.3)exists a unique positive solution ( u 1 (t), u 2 (t)) and which is globally asymptotically stable, where Thus, there exists a positive constant η 1 small enough for the above 1 , such that u By the comparison principle, there exists t 2 ≥ t 1 for the above From the second, fifth, seventh and tenth equations of model (2.1), we have Considering the following auxiliary system where Z = (z 1 , z 2 ) T .From Lemma 2.1, system (4.4) has a positive T-periodic solution Z(t) such that Z(t) = Z(t)e µ 1 t , where In the following, we prove that lim inf t→+∞ I i (t) ≥ η, i = h, m.From the above discussion, we consider only the following two possibilities: (ii) I i (t) oscillates about η for all large t, i = h, m.
If case (i) holds, then we have completely the result.Next, we turn to case (ii).Owing to lim sup t→+∞ I i (t) ≥ η, there exists a Integrating the above equation from t 1 to t, we get Moreover, from the fifth and tenth equations of model (2.1), it follows that And, integrating the equation from t 1 to t, we have For t > t 2 , the similar arguments can be continued and we similarly get non-infinitesimal positive η 2 .Therefore, we can get the sequence {η k }, where Hence from the above discuss, we get I i (t) ≥ η > 0, i = h, m for all t ≥ t 1 .The proof is complete.

Forward and backward bifurcation of endemic periodic solutions
In this section, we proceed to study bifurcation using the bifurcation theory (more details can be found in Lakmeche et al. [22]).Let the culling rate φ be the bifurcation parameter.We define the solution vector X(t) := (S h (t), S m (t), R h (t), S m (t), I m (t)), the mapping F(X(t)) = (F 1 (X(t)), . . ., F 5 (X(t))) : R 5 → R 5 by the right hand side of the first to fifth equations of model (2.1), and the mapping Furthermore, we define Φ(t, X 0 ), 0 < t ≤ T, to be the solution of model consisting of the first to fifth equations of model (2.1), where X 0 = X(0).Then X(T) = Φ(T, X 0 ) := Φ(X 0 ) and X(T + ) = I(φ, Φ(X 0 )).We define the operator Ψ by Ψ(φ, X) := I(φ, Φ(X)), where Ψ(φ, X) = (Ψ 1 (φ, X), . . ., Ψ 5 (φ, X)).Denote D X Ψ the derivative of Ψ with respect to X. Then X is a periodic solution of period T for model (2.1) if and only if its initial value X 0 is a fixed point for Ψ(φ, X).Namely, Ψ(φ, X 0 ) = X 0 .Consequently, to establish the existence of nontrivial periodic solutions of model (2.1), one needs to prove the existence of the nontrivial fixed point of Ψ.
We know that the threshold value R 0 decreases as φ increase.Then a supercritical bifurcation means a backward bifurcation in the model while the subcritical bifurcation equated to a forward bifurcation in the φ − α 1 plane.Thus we have the following theorem.Theorem 5.2.As the parameter φ passes through the critical value φ 0 , a backward bifurcation occurs if BC < 0, or else there will be a forward bifurcation as BC > 0 at R 0 = 1.

Numerical simulation and discussion
In this paper, a mathematical model of dengue fever with impulsive culling of mosquitoes, saturation and bilinear incidence are considered.The main purpose is to investigate the effect of impulsive culling mosquitoes strategy, which govern whether the dengue fever dies out or not, and further to examine how the impulsive culling control strategy affects the prevention and control of dengue fever.By using the comparison principle, integral and differential inequalities, the way of spectral radius and analytical methods, some sufficient conditions for the existence and stability of disease-free periodic solution, and the uniform persistence of disease are obtained.Theoretical results show that dengue fever can be controlled via adjusting the control parameters of the model that depend on these conditions.
We choose, firstly, culling period T = 6 (days) and culling rate φ = 0.72.It is easy to calculate that R 0 ≈ 0.9385 < 1.So, from Theorem 3.2, we obtain that model (2.1) has a disease-free periodic solution which is globally asymptotically stable.The quantities of infectious human, infectious mosquitoes and uninfected mosquitoes in model (2.1) with or without impulsive culling are plotted against time in Figure 6.1 (a)-(c) with blue lines and red lines, respectively.Infected mosquitoes and infectious human in model (2.1) with impulsive culling control strategy and the stability of disease-free periodic solution of model (2.1) with or without impulsive culling are plotted in Figure 6.1 (d) with blue lines and red lines, respectively.Theoretical results and numerical simulations imply that we can eliminate dengue fever through vector-control strategy.
Secondly, we choose T = 10 (days), φ = 0.2 and others parameters are fixed as above.By calculating, we get R 0 ≈ 4.7336 > 1.Therefore, model (2.1) has a positive periodic solution from Theorem 4.1.Figures 6.2 (a)-(c) show that the numerical solutions of infectious human, infectious mosquitoes and uninfected mosquitoes with different initial values.The plots in Figure 6.2 (d) show the uniform persistence of infected mosquitoes and infected humans.Theoretical results and numerical simulations show that dengue fever is uniformly persistent if the culling strength φ is low and the culling cycle period T is not too long.
Thirdly, we consider the frequencies of culling and culling rate how to impact on the uniform persistence and extinction of disease.We fixed φ = 0.1 and other parameters are invariant, and chose different control period T for model (2.1).Figures 6.4 (a) and 6.4 (b) show the quantities of infectious human and infected mosquitoes with T = 1, 7 and 16 (days), respectively.Numerical simulations imply that disease is extinct when culling period T is short and disease is uniform persistent for long culling period T. Further, we fixed T = 3 (days), and chose culling rate φ = 0.1, 0.4 and 0.8 for model (2.1), respectively.The plots in Figures 6.4 (c) and 6.4 (d) show the quantities of infectious human and infected mosquitoes, which imply that the disease is extinct for high culling rate φ and is uniform persistent for low culling rate.All of these simulations indicate that the high frequency of culling and large culling rate are necessary for the goal of eliminating disease, The bifurcation diagram of infectious human and infected mosquitoes about culling rate φ in Figure 6.3 also accord with this result.
Finally, it is important to emphasize that the factors of seasonal variation in mosquito population size, the latent period of infected mosquitoes, the dispersion of both humans and mosquitoes, and vertical transmission of the virus in the mosquito population, affect the dynamical behaviors of both mosquitoes and humans and hence disease spread between mosquitoes and humans.We leave these topics for future work.
2), we further get Thus we obtain It is obvious that Consequently, we have d dt We solve the above equations and denote Appendix B The first-order partial derivatives of N 5 (φ, α 1 ) We can easily get From the equation of (5.6) as i = 1, we have Since Y 1 is a basis in Ker(D X N(0, 0)), then we have Similarly, from the equation of (5.6) as i = 2, 3, 4, we can obtain that It is obvious from (B.4) and (B.5) that Considering equation (5.1) as i = 1, we have with X 0 = (X 01 , X 02 , X 03 , X 04 , X 05 ), and X = (X 1 , X 2 , X 3 , X 4 , X 5 ).Thus one obtains We can similarly obtain from (5.1) as i = 2, 3, 4 that 0 = ∂N 2 (0, 0) ∂φ From equations (B.8) and (B.9), we get we can thus observe from (B.1), (B.3), (B.6) and (B.10) that Appendix C The second-order partial derivatives of N 5 (φ, α 1 ) with respect to φ From equation (5.3), we have Then d dt It is obvious that We can similarly obtain that d dt Consider the initial conditions it can be deduced from (C.1) and (C.2) that The same method can be adopted to get that Based on the third equation of (B.9), we have Substituting (B.10) and (C.3) into the above equation, we have Therefore, we can easily get that We will first calculate the value of Once again, by substituting (B.10), (C.2) and (C.4) into the above equation, we can thus deduce that Substituting (D.1) and (B.9) into (D.2), we can easily get Appendix E The second-order partial derivatives of N 5 (φ, α 1 ) with respect to α 1 By calculating we have
Then we will consider the solution of the following equation from model (2.1) in the time interval [t 1 , t 2 ].From the second equation of model (2.1), one have dI h (t) dt ≥ −(µ h + d h + γ)I h (t).